Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.07218 · Transportation Research Part C Emerging Technologies

Improving Feasibility in Quantum Approximate Optimization Algorithm for Vehicle Routing via Constraint-Aware Initialization and Hybrid XY-X Mixing

Yuan-Zheng Lei, Yaobang Gong, Xianfeng Terry Yang, Nii Attoh-Okine

cs.ETquant-phSubmitted Apr 8, 2026• Announced Apr 9, 2026• 0 citations• Revised Sep 30, 2026 (v2)
v1 · Apr 8, 2026v2 · Sep 30, 2026
arXivHTMLPDF

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading framework for quantum combinatorial optimization. The Vehicle Routing Problem (VRP), a core problem in logistics and transportation, is a natural application target, but it poses a major feasibility challenge for standard QAOA because feasible solutions occupy only a tiny fraction of the search space, and the conventional Pauli-$X$ mixer can disrupt partial solution structures that satisfy key local constraints. To address this issue, we propose a constraint-aware QAOA framework with two complementary components. First, we design a lightweight initialization strategy that encodes a selected subset of simple yet informative local one-hot constraints into the initial state, thereby reducing the initial superposition space and increasing the probability mass on states with important local structure. Second, we introduce a hybrid XY-$X$ mixer that preserves the constraint structure imposed at initialization while retaining exploratory flexibility over the remaining unconstrained degrees of freedom during QAOA evolution. We evaluate the proposed framework against standard QAOA under three progressively more realistic regimes: ideal statevector simulation, finite-shot sampling, and noisy finite-shot sampling. Across all regimes, the proposed method consistently achieves lower average energy and higher feasible-solution ratios than standard QAOA, indicating more effective guidance toward structurally valid, lower-cost VRP solutions. However, the performance gap narrows in the noisy regime. Because this setting adopts a hardware-inspired error model based on near-best-reported laboratory-level qubit gate and readout fidelities, the observed attenuation suggests that the practical advantage of the more structured mixer is likely to grow as quantum hardware improves and error rates decline.

Journal reference
Transportation Research Part C: Emerging Technologies, 194, 106047 (2027)
DOI
10.1016/j.trc.2026.106047

Related papers

Feasibility-Preserving Quantum Search for Constrained Transportation Routing

Dahye Kim, Monika Filipovska

Transportation routing problems such as the Traveling Salesperson Problem (TSP) and the Vehicle Routing Problem (VRP) are characterized by strict feasibility requirements involving customer assignment and visit rules, route sequencing, and depot-return logic alongside cost minimization. Most quantum routing formulations adopt Quadratic Unconstrained Binary Optimization (QUBO) encodings, where feasibility is incorporated indirectly via penalty terms in the cost Hamiltonian. While convenient for standard implementations of the Quantum Approximate Optimization Algorithm (QAOA), QUBO encodings allow the quantum search dynamics to allocate substantial probability to infeasible route configurations. This study develops a transportation-grounded constraint-aware Quantum Alternating Operator Ansatz (QAOA+) framework that embeds feasibility-preserving logic directly into the search operator. We introduce a custom mixer that functions as a quantum analogue of feasibility-preserving routing neighborhoods, using column-wise swap moves, it restricts evolution to feasible configurations while enabling structured exploration of valid routes. We compare three constraint-handling architectures: penalty-based QUBO QAOA, penalty free QAOA+ with the feasibility-preserving mixer, and a Hybrid QAOA+ combining mixer based feasibility with and penalty guidance. Results on small TSP and VRP instances show that constraint-handling architecture strongly influences feasible-route sampling, convergence behavior, and probability concentration over low-cost feasible routes. These findings position constraint-aware quantum search as a methodological extension of transportation routing search approaches, where feasibility is enforced through admissible quantum transitions rather than post-hoc penalties.

quant-pharXiv:2608.05394•Aug 7, 2026•0 citations

Warm-Starting QAOA With XY Mixers: A Novel Approach for Traveling Salesman Problem

Rafael Simões do Carmo, Marcos Cleison Silva Santana, F. F. Fanchini +2 more

Quantum optimization algorithms, such as the Quantum Approximate Optimization Algorithm, are emerging as promising heuristics for solving complex combinatorial problems. To improve performance, several extensions to the standard QAOA framework have been proposed in recent years. Two notable directions include: warm-starting techniques, which incorporate classical approximate solutions to guide the quantum evolution, and custom mixer Hamiltonians, such as XY mixers, which constrain the search to feasible subspaces aligned with the structure of the problem. In this work, we propose an approach that integrates these two strategies: a warm-start initialization with an XY mixer ansatz, enabling constraint-preserving quantum evolution biased toward high-quality classical solutions. The method begins by reformulating the combinatorial problem as a MaxCut instance, solved approximately using the Goemans-Williamson algorithm. The resulting binary solution is relaxed and used to construct a biased superposition over valid one-hot quantum states, maintaining compatibility with the XY mixer's constraints. We evaluate the approach on 5-city instances of the Traveling Salesperson Problem, a canonical optimization problem frequently encountered as a subroutine in real-world Vehicle Routing Problems. Our method is benchmarked against both the standard XY-mixer QAOA and a warm-start-only variant based on MaxCut relaxation. Results show that the proposed combination consistently outperforms both baselines in terms of the percentage and rank of optimal solutions, demonstrating the effectiveness of combining structured initializations with constraint-aware quantum evolution for optimization problems.

arXiv:2504.19934•IEEE Transactions on Intelligent Transportation Systems•Jan 1, 2026•0 citations

A feasibility-preserved quantum approximate solver for the Capacitated Vehicle Routing Problem

Ningyi Xie, Xinwei Lee, Dongsheng Cai +3 more

Abstract The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that arises in various fields including transportation and logistics. The CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the most efficient plan for a fleet of vehicles to deliver goods to a set of customers, subject to the limited carrying capacity of each vehicle. As the number of possible solutions skyrockets when the number of customers increases, finding the optimal solution remains a significant challenge. Recently, the Quantum Approximate Optimization Algorithm (QAOA), a quantum-classical hybrid algorithm, has exhibited enhanced performance in certain combinatorial optimization problems compared to classical heuristics. However, its ability diminishes notably in solving constrained optimization problems including the CVRP. This limitation primarily arises from the typical approach of encoding the given problems as penalty-inclusive binary optimization problems. In this case, the QAOA faces challenges in sampling solutions satisfying all constraints. Addressing this, our work presents a new binary encoding for the CVRP, with an alternative objective function of minimizing the shortest path that bypasses the vehicle capacity constraint of the CVRP. The search space is further restricted by the constraint-preserving mixing operation. We examine and discuss the effectiveness of the proposed encoding under the framework of the variant of the QAOA, Quantum Alternating Operator Ansatz (AOA), through its application to several illustrative examples. Compared to the typical QAOA approach, the proposed method not only preserves the feasibility but also achieves a significant enhancement in the probability of measuring optimal solutions.

arXiv:2308.08785•Quantum Information Processing•Jul 29, 2024•24 citations

Improving QAOA feasibility and hardware performance for vehicle routing via linear chain and ramp schedules

Talha Azfar, Ruimin Ke

The Quantum Approximate Optimization Algorithm (QAOA) is difficult to deploy on near-term hardware for constrained routing problems because feasibility constraints produce dense interaction graphs, deep transpiled circuits, and costly parameter search. We study this challenge for Vehicle Routing Problem (VRP) instances using a hardware-aware QAOA workflow that combines linear-ramp parameter initialization, Conditional Value-at-Risk (CVaR) optimization, and a Linear-Chain QAOA (LC-QAOA) surrogate that retains local ZZ interactions to reduce two-qubit depth. A matched full-QAOA versus LC-QAOA feasibility comparison is performed for 4-node instances, while the 5- and 6-node experiments characterize circuit executability and low-probability optimum recovery at the tested depths. In 4-node hardware runs, LC-QAOA increases mean feasible-sample probability from 0.182% to 0.917% and mean optimum-sample probability from 0.045% to 0.239%, giving approximately \(5\times\) improvement over full QAOA. At 6 nodes, a full-QAOA circuit at \(p=1\) already has large transpiled two-qubit depth, while LC-QAOA remains executable and samples CPLEX-certified optima in selected hardware runs at depths p =10–13, albeit with low optimum probability. Overall, the results identify LC-QAOA as a hardware-efficient surrogate heuristic that trades Hamiltonian completeness for lower depth and improved feasible-solution sampling on noisy quantum hardware.

Scientific Reports•Sep 10, 2026•0 citations

Constrained Quantum Optimization via Iterative Warm-Start XY-Mixers

David Bucher, Maximilian Janetschek, Michael Poppel +3 more

The Quantum Approximate Optimization Algorithm (QAOA) is a leading hybrid heuristic for combinatorial optimization, but efficiently handling hard constraints remains a significant challenge. XY-mixers successfully confine quantum state evolution to a feasible subspace, such as the Hamming-weight-1 sector for one-hot constraints. On the contrary, warm-starting biases the search toward promising regions based on preliminary solutions. Combining these two techniques requires maintaining the essential alignment between the initial state and the mixer Hamiltonian to preserve convergence guarantees. Previous work demonstrated warm-starting with XY-mixers via a biased initial state, but relying only on standard mixer Hamiltonians. Consequently, the initial state is no longer a ground state of the mixer. In this work, we overcome these limitations by formulating a warm-started XY-mixer Hamiltonian for one-hot constraints and proving its ground-state properties. Furthermore, we provide a shallow circuit implementation suitable for NISQ implementations. We embed the warm-starting into a classical heuristic that iteratively updates the bias based on previous samples, called Iterative Warm-Starting (IWS). Extensive numerical simulations on Max-$k$-Cut and Traveling Salesperson Problem instances demonstrate that IWS-QAOA significantly accelerates the solution-finding process, increasing the probability of sampling optimal solutions by orders of magnitude compared to standard XY-QAOA. Finally, we validate our approach on the ibm_boston QPU using hardware-tailored 144-qubit problem instances. By coupling IWS-QAOA with a greedy steepest-descent post-processing strategy to repair infeasible measurements caused by hardware noise, we successfully identify optimal solutions on actual quantum devices.

quant-pharXiv:2604.02083•New Journal of Physics•Apr 3, 2026•0 citations

Quantum-Assisted Vehicle Routing: Realizing QAOA-based Approach on Gate-Based Quantum Computer

Talha Azfar, Osama M. Raisuddin, Ruimin Ke +1 more

The Vehicle Routing Problem (VRP) is a fundamental combinatorial optimization challenge with broad applications in logistics and transportation. In this work, we present a quantum-assisted framework that integrates the Quantum Approximate Optimization Algorithm (QAOA) with a link-based formulation of VRP. Our approach encodes flow conservation and subtour elimination directly into the cost Hamiltonian, preserving graph structure while minimizing resource requirements for practical hardware implementation. We design and implement the full pipeline on a gate-based quantum computer, including problem formulation, encoding, circuit synthesis, and execution on IBM Quantum System One. Experimental results on small VRP instances highlight the effects of penalty scaling, coefficient normalization, and circuit depth on solution feasibility under hardware noise. While scalability remains constrained by circuit complexity and decoherence, the study demonstrates a practical pathway for implementing VRP on quantum hardware and identifies methodological directions for advancing near-term quantum optimization.

arXiv:2505.01614•May 2, 2025•3 citations
The Quantum Approximate Optimization Algorithm (QAOA) is a leading framework for quantum combinatorial optimization. The Vehicle Routing Problem (VRP), a core problem in logistics and transportation, is a natural application target, but it poses a major feasibility challenge for standard QAOA because feasible solutions occupy only a tiny fraction of the search space, and the conventional Pauli-XXX mixer can disrupt partial solution structures that satisfy key local constraints. To address this issue, we propose a constraint-aware QAOA framework with two complementary components. First, we design a lightweight initialization strategy that encodes a selected subset of simple yet informative local one-hot constraints into the initial state, thereby reducing the initial superposition space and increasing the probability mass on states with important local structure. Second, we introduce a hybrid XY-XXX mixer that preserves the constraint structure imposed at initialization while retaining exploratory flexibility over the remaining unconstrained degrees of freedom during QAOA evolution. We evaluate the proposed framework against standard QAOA under three progressively more realistic regimes: ideal statevector simulation, finite-shot sampling, and noisy finite-shot sampling. Across all regimes, the proposed method consistently achieves lower average energy and higher feasible-solution ratios than standard QAOA, indicating more effective guidance toward structurally valid, lower-cost VRP solutions. However, the performance gap narrows in the noisy regime. Because this setting adopts a hardware-inspired error model based on near-best-reported laboratory-level qubit gate and readout fidelities, the observed attenuation suggests that the practical advantage of the more structured mixer is likely to grow as quantum hardware improves and error rates decline.
Transportation routing problems such as the Traveling Salesperson Problem (TSP) and the Vehicle Routing Problem (VRP) are characterized by strict feasibility requirements involving customer assignment and visit rules, route sequencing, and depot-return logic alongside cost minimization. Most quantum routing formulations adopt Quadratic Unconstrained Binary Optimization (QUBO) encodings, where feasibility is incorporated indirectly via penalty terms in the cost Hamiltonian. While convenient for standard implementations of the Quantum Approximate Optimization Algorithm (QAOA), QUBO encodings allow the quantum search dynamics to allocate substantial probability to infeasible route configurations. This study develops a transportation-grounded constraint-aware Quantum Alternating Operator Ansatz (QAOA+) framework that embeds feasibility-preserving logic directly into the search operator. We introduce a custom mixer that functions as a quantum analogue of feasibility-preserving routing neighborhoods, using column-wise swap moves, it restricts evolution to feasible configurations while enabling structured exploration of valid routes. We compare three constraint-handling architectures: penalty-based QUBO QAOA, penalty free QAOA+ with the feasibility-preserving mixer, and a Hybrid QAOA+ combining mixer based feasibility with and penalty guidance. Results on small TSP and VRP instances show that constraint-handling architecture strongly influences feasible-route sampling, convergence behavior, and probability concentration over low-cost feasible routes. These findings position constraint-aware quantum search as a methodological extension of transportation routing search approaches, where feasibility is enforced through admissible quantum transitions rather than post-hoc penalties.
Quantum optimization algorithms, such as the Quantum Approximate Optimization Algorithm, are emerging as promising heuristics for solving complex combinatorial problems. To improve performance, several extensions to the standard QAOA framework have been proposed in recent years. Two notable directions include: warm-starting techniques, which incorporate classical approximate solutions to guide the quantum evolution, and custom mixer Hamiltonians, such as XY mixers, which constrain the search to feasible subspaces aligned with the structure of the problem. In this work, we propose an approach that integrates these two strategies: a warm-start initialization with an XY mixer ansatz, enabling constraint-preserving quantum evolution biased toward high-quality classical solutions. The method begins by reformulating the combinatorial problem as a MaxCut instance, solved approximately using the Goemans-Williamson algorithm. The resulting binary solution is relaxed and used to construct a biased superposition over valid one-hot quantum states, maintaining compatibility with the XY mixer's constraints. We evaluate the approach on 5-city instances of the Traveling Salesperson Problem, a canonical optimization problem frequently encountered as a subroutine in real-world Vehicle Routing Problems. Our method is benchmarked against both the standard XY-mixer QAOA and a warm-start-only variant based on MaxCut relaxation. Results show that the proposed combination consistently outperforms both baselines in terms of the percentage and rank of optimal solutions, demonstrating the effectiveness of combining structured initializations with constraint-aware quantum evolution for optimization problems.
Abstract The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that arises in various fields including transportation and logistics. The CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the most efficient plan for a fleet of vehicles to deliver goods to a set of customers, subject to the limited carrying capacity of each vehicle. As the number of possible solutions skyrockets when the number of customers increases, finding the optimal solution remains a significant challenge. Recently, the Quantum Approximate Optimization Algorithm (QAOA), a quantum-classical hybrid algorithm, has exhibited enhanced performance in certain combinatorial optimization problems compared to classical heuristics. However, its ability diminishes notably in solving constrained optimization problems including the CVRP. This limitation primarily arises from the typical approach of encoding the given problems as penalty-inclusive binary optimization problems. In this case, the QAOA faces challenges in sampling solutions satisfying all constraints. Addressing this, our work presents a new binary encoding for the CVRP, with an alternative objective function of minimizing the shortest path that bypasses the vehicle capacity constraint of the CVRP. The search space is further restricted by the constraint-preserving mixing operation. We examine and discuss the effectiveness of the proposed encoding under the framework of the variant of the QAOA, Quantum Alternating Operator Ansatz (AOA), through its application to several illustrative examples. Compared to the typical QAOA approach, the proposed method not only preserves the feasibility but also achieves a significant enhancement in the probability of measuring optimal solutions.
The Quantum Approximate Optimization Algorithm (QAOA) is difficult to deploy on near-term hardware for constrained routing problems because feasibility constraints produce dense interaction graphs, deep transpiled circuits, and costly parameter search. We study this challenge for Vehicle Routing Problem (VRP) instances using a hardware-aware QAOA workflow that combines linear-ramp parameter initialization, Conditional Value-at-Risk (CVaR) optimization, and a Linear-Chain QAOA (LC-QAOA) surrogate that retains local ZZ interactions to reduce two-qubit depth. A matched full-QAOA versus LC-QAOA feasibility comparison is performed for 4-node instances, while the 5- and 6-node experiments characterize circuit executability and low-probability optimum recovery at the tested depths. In 4-node hardware runs, LC-QAOA increases mean feasible-sample probability from 0.182% to 0.917% and mean optimum-sample probability from 0.045% to 0.239%, giving approximately 5×5\times5× improvement over full QAOA. At 6 nodes, a full-QAOA circuit at p=1p=1p=1 already has large transpiled two-qubit depth, while LC-QAOA remains executable and samples CPLEX-certified optima in selected hardware runs at depths p =10–13, albeit with low optimum probability. Overall, the results identify LC-QAOA as a hardware-efficient surrogate heuristic that trades Hamiltonian completeness for lower depth and improved feasible-solution sampling on noisy quantum hardware.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading hybrid heuristic for combinatorial optimization, but efficiently handling hard constraints remains a significant challenge. XY-mixers successfully confine quantum state evolution to a feasible subspace, such as the Hamming-weight-1 sector for one-hot constraints. On the contrary, warm-starting biases the search toward promising regions based on preliminary solutions. Combining these two techniques requires maintaining the essential alignment between the initial state and the mixer Hamiltonian to preserve convergence guarantees. Previous work demonstrated warm-starting with XY-mixers via a biased initial state, but relying only on standard mixer Hamiltonians. Consequently, the initial state is no longer a ground state of the mixer. In this work, we overcome these limitations by formulating a warm-started XY-mixer Hamiltonian for one-hot constraints and proving its ground-state properties. Furthermore, we provide a shallow circuit implementation suitable for NISQ implementations. We embed the warm-starting into a classical heuristic that iteratively updates the bias based on previous samples, called Iterative Warm-Starting (IWS). Extensive numerical simulations on Max-kkk-Cut and Traveling Salesperson Problem instances demonstrate that IWS-QAOA significantly accelerates the solution-finding process, increasing the probability of sampling optimal solutions by orders of magnitude compared to standard XY-QAOA. Finally, we validate our approach on the ibm_boston QPU using hardware-tailored 144-qubit problem instances. By coupling IWS-QAOA with a greedy steepest-descent post-processing strategy to repair infeasible measurements caused by hardware noise, we successfully identify optimal solutions on actual quantum devices.
The Vehicle Routing Problem (VRP) is a fundamental combinatorial optimization challenge with broad applications in logistics and transportation. In this work, we present a quantum-assisted framework that integrates the Quantum Approximate Optimization Algorithm (QAOA) with a link-based formulation of VRP. Our approach encodes flow conservation and subtour elimination directly into the cost Hamiltonian, preserving graph structure while minimizing resource requirements for practical hardware implementation. We design and implement the full pipeline on a gate-based quantum computer, including problem formulation, encoding, circuit synthesis, and execution on IBM Quantum System One. Experimental results on small VRP instances highlight the effects of penalty scaling, coefficient normalization, and circuit depth on solution feasibility under hardware noise. While scalability remains constrained by circuit complexity and decoherence, the study demonstrates a practical pathway for implementing VRP on quantum hardware and identifies methodological directions for advancing near-term quantum optimization.